Upper join-closed subgroup property

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This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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This article is about a general term. A list of important particular cases (instances) is available at Category:Upper join-closed subgroup properties

Definition

Definition with symbols

A subgroup property p is said to be upper join-closed if given H \le G and K_i, i \in I are intermediate subgroups of G containing H (indexed by a nonempty set I) and H satisfies p in each K_i, we have that H satisfies p in the join of subgroups \langle K_i \rangle_{i \in I}.

Relation with other metaproperties

Stronger metaproperties

Weaker metaproperties

Related notions

Given a subgroup property p that is identity-true, upper join-closed and also satisfies the intermediate subgroup condition, we can, given any subgroup H of G associate a unique largest subgroup M containing H for which H satisfies p in M.

Such a subgroup property is termed an izable subgroup property and the M that we get is termed the izing subgroup of H for that subgroup property.

Properties satisfying it

Normality

Normality is an upper join-closed subgroup property, viz, if H \le G and K_1, K_2 are intermediate subgroups such that H \triangleleft K_1 and H \triangleleft K_2, then H \triangleleft <K_1,K_2>.

Central factor

The property of being a central factor is also upper join-closed, in fact, it is izable.